How to prove that $\lim\limits_{n\to\infty}\left(1+\frac{1}{p_n}\right)^{p_n}=e$?

Question

Suppose that $\{p_n\}$ is an arbitrary positive sequence that $\lim\limits_{n\to\infty} p_n=+\infty$. How to prove that $\lim\limits_{n\to\infty}\left(1+\dfrac{1}{p_n}\right)^{p_n}=e$?

Answer 1

Write that $$ \left(1+\frac{1}{p_n}\right)^{p_n}=e^{p_n\ln\left(1+\frac{1}{p_n}\right)} $$ And $1/p_n \underset{n \rightarrow +\infty}{\rightarrow}0 \ $ so $$\ln\left(1+\frac{1}{p_n}\right)=\frac{1}{p_n}+o\left(\frac{1}{p_n}\right)$$ And finally

$$ e^{p_n\ln\left(1+\frac{1}{p_n}\right)}=e^{1+o\left(1\right)}\underset{n \rightarrow +\infty}{\rightarrow}e $$